| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 2·11-s − 4·13-s + 15-s + 6·17-s − 6·19-s − 21-s − 8·23-s + 25-s + 27-s + 2·29-s + 10·31-s − 2·33-s − 35-s − 2·37-s − 4·39-s + 10·41-s + 4·43-s + 45-s − 8·47-s + 49-s + 6·51-s − 4·53-s − 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.258·15-s + 1.45·17-s − 1.37·19-s − 0.218·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.79·31-s − 0.348·33-s − 0.169·35-s − 0.328·37-s − 0.640·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.840·51-s − 0.549·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.924360445216681810381081668167, −6.96921426855486686542861513994, −6.18441126514015494265739802506, −5.62329287202880070434024050020, −4.63874769353103061633113450424, −4.05441784975223583692100422406, −2.87733927036303907937070944462, −2.52403655450759303446957333049, −1.45034079784553119415967492287, 0,
1.45034079784553119415967492287, 2.52403655450759303446957333049, 2.87733927036303907937070944462, 4.05441784975223583692100422406, 4.63874769353103061633113450424, 5.62329287202880070434024050020, 6.18441126514015494265739802506, 6.96921426855486686542861513994, 7.924360445216681810381081668167
